Is there a difference between the order type of Q·ω and Q·Q?

Question

From what I understand, the expression is "a countable amount of the order type of Q", which intuitively should be equal to the second expression. Is this true? How do I explain this formally?

Thanks for your time.

Answer 1

Technically, there is a difference. Because we take a lexicographic order on two different products. One which has a discrete order and another which has a dense order.

As luck would have, however, both these products are dense everywhere and without endpoints, and the following theorem finishes the work:

Theorem. (Cantor) Suppose that $(A,\leq_A)$ and $(B,\leq_B)$ are two countable dense linear orders without endpoints. Then $(A,\leq_A)$ is order isomorphic to $(B,\leq_B)$.

So while technically these are different, they are still isomorphic, so they have the same order type. That of $\Bbb Q$.